# Research

## Discrete Mathematics

### Monday, April 24, 2017

Title: Sequential tensor categories and higher Picard groups 2
Speaker: El-kaïoum Moutuou
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

This time I will introduce the concept of $$n$$-equential categories, define higher Picard groups for objects is such categories. Then I will try hard to convince you these groups are quite interesting, especially by showing to which extent they can provide a geometric picture of lower degree cohomology groups of topological spaces.

### Monday, April 17, 2017

Title: TBA
Speaker: Lindsay Fields
Time: 2:00pm‐3:00pm
Place: CMC 108

TBA

### Monday, April 10, 2017

Title: The Reed-Muller Code in Finite Space $$F_q$$
Speaker: Hongliang Wang
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

I am going to introduce the one of many ways to construct of Reed-Muller code $$R_q(r,n)$$, which was first discovered as a binary code by Muller and Reed in 1954. $$R_q(r,n)$$ is an important linear error-correcting code and has a lot of applications in engineering. And I will also introduce some properties of this code such as dimension and minimum distance.

### Monday, April 3, 2017

Title: Algorithms for detecting critical parameters of random graph models
Speaker: Razvan Teodorescu
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

In 2000, Schramm and collaborators showed that the loop-erased random walk generation of uniform spanning trees in an infinite-size graph (the Wilson algorithm) is independent on boundary conditions under certain conditions satisfied by the discrete Laplace operator on that graph. Furthermore, this can be used to determine the critical dimension of a random lattice, at which the number of components of the uniform spanning trees becomes 1. We investigate a generalization of this approach, for other geometric characteristics of the random graph, and identify the corresponding process whose scaling behavior can identify the critical parameters.

Joint work with I. Teodorescu and P. Warman.

### Monday, March 27, 2017

Title: Zassenhaus With or Without Cohomology
Speaker: Greg McColm
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

One of the major results of Hans Zassenhaus is that if a group has a maximal abelian subgroup that is (1) normal, (2) free, (3) finitely generated, and (4) of finite index, then that group is crystallographic. (The converse is also true, and a consequence of the Bieberbach theorems.) Modern expositions of Zassenhaus' Theorem tend to rely on the cohomology of groups.

In this presentation, we will look at this result from a crystallographic (geometric) point of view, and then we will outline the proof from Schwarzenberger, sans cohomology. We will conclude with a discussion of how cohomology fits into this proof.

This presentation does not presume any familiarity with cohomology.

### Monday, March 20, 2017

Title: The Eilenberg-Moore Spectral Sequence and applications to the Cohomology of Loop of Suspended Spaces
Speaker: Emanuele Zappala
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

In this talk I will give a brief introduction to Spectral Sequences and I will focus in particular on the Eilenberg-Moore Spectral Sequence. Spectral Sequences are a powerful Algebraic tool to compute Homologies and Cohomologies of certain spaces.

I will present two main results due to Eilenberg and Moore, one of them completely Algebraic and the second one Topological, whose combination gives a good way to tackle the problem of determining the Cohomology of a Loop Suspended Space (and in principle the iterated process of applying the Suspension Functor and the Loop Functor). General results in this direction have been discovered many years ago using a very Geometric approach (Moore Loops, James Construction, Adams-Hilton Construction, Adams Cobar Construction, Zilchgon models, Boardman's Little Cubes, May-Milgram Configuration Spaces), I would like to give a different point of view, almost completely Algebraic.

After a brief digression on Steenrod Operations, time permitting, I would like to prove that the Cohomology graded Algebra of a single Loop of a singly Suspended Space is Polynomial whenever the base space ha no nilpotents.

### Monday, March 13, 2017

Spring Break — no seminar this week.

### Monday, March 6, 2017

Title: A generalization of the reversed Dickson polynomials over finite fields
Speaker: Neranga Fernando, Northeastern University
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

Let $$p$$ be a prime and $$q$$ a power of $$p$$. Let $$\mathbb{F}_q$$ be the finite field with $$q$$ elements. Permutation polynomials over finite fields have important applications in coding theory, cryptography, finite geometry, combinatorics and computer science, among other fields. Recently, reversed Dickson polynomials over finite fields have been studied extensively by many for their general properties and permutation behaviour.

For $$a\in\mathbb{F}_q$$, the $$n$$-th reversed Dickson polynomial of the $$(k+1)$$-th kind $$D_{n,k}(a,x)$$ is defined by $$D_{n,k}(a,x)=\sum_{i=0}^{\lfloor\frac n2\rfloor}\frac{n-ki}{n-i}\dbinom{n-i}{i}(-x)^{i}a^{n-2i},$$ and $$D_{0,k}(a,x)=2-k$$.

I am primarily interested in the question: When is $$D_{n,k}(a,x)$$ a permutation polynomial (PP) of $$\mathbb F_q$$? In this talk, I will completely explain the permutation behaviour of the reversed Dickson polynomials of the $$(k+1)$$-th kind $$D_{n,k}(a,x)$$ when $$a=0$$, $$n=p^l$$, $$n=p^l+1$$, and $$n=p^l+2$$, where $$l\ge 0$$ is an integer. I will also talk about some general properties of the reversed Dickson polynomials of the $$(k+1)$$-th kind.

In particular, I will explain the explicit evaluation of the sum $$\sum\limits_{a\in f_q}D_{n,k}(1,a)$$ which provides a necessary condition for $$D_{n,k}(1,x)$$ to be a PP of $$\mathbb{F}_q$$. These results unify and generalize numerous recently discovered results on reversed Dickson polynomials over finite fields.

### Monday, February 27, 2017

Title: A Model for Sensory Representation: Mapping of Discrete Sensor Arrays onto Neural Networks and the Cortical Brain
Speaker: Apurva Bhatty
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

Sensors may be arrayed regularly or irregularly with a particular embedding onto the environment or onto an organism's body. The sensors may all be of the same type or of multiple types. The connections from the sensors onto the “processing apparatus” may be distinct for each type of sensor (“labelled lines”), or they may be indistinguishable (“unlabelled lines”).

I will present a simple model of sensory information processing as mappings between low dimensional manifolds, and show that this can represent “nonsymbolic” information and “symbolic” information in the same family of neural circuits. I will review how each of these distinctions affects the possible higher level representations of the sensors based upon this model.

### Monday, February 20, 2017

Title: The Amorphous Halting Problem
Speaker: Richard Stark
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

Turing viewed Life as a most extreme form of information processing. He idealized it as morphogenesis, especially as seen in the paradox of leopards' spots. This paradox was Turing's last great problem. He resolved it in The Chemical Basis of Morphogenesis, 1952. The key was to view diffusion as a form of information processing.

Decades later, TCBM and diffusion became a part of the study of natural information processing. In the 1990s, Hal Abelson, et al. used diffusion in his Amorphous Processes (morphogenesis has none of the precise structure seen in man-made computers, so it is said to be amorphous). Abelson gave an engineer's definition of this biological form of information processing. At the same time, I (being a student of Kleene's) developed a mathematical formalism for amorphous processes equivalent to Abelson's informal definition.

Continuing along the lines of my talk last spring, e.g., from the set of all connected simple graphs, the bipartite graphs are amorphously computable, I will use my formalism to demonstrate a property of amorphous computations which guarantees that the not-bipartite graphs are not amorphously computable.

Finally, concluding that the Amorphous Halting Problem is not amorphously solvable.

### Monday, February 13, 2017

Title: Sequential tensor categories and higher Picard groups
Speaker: El-kaïoum Moutuou
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

I will talk about tensor categories, introduce the concept of $$n$$-sequential categories, define higher Picard groups for objects is such categories. Then, if time allows, I will try to convince you these groups are quite interesting by showing how, when defined in an appropriate category, they are related to the lower degree cohomology groups of topological spaces.

### Monday, February 6, 2017

Title: Factors in graphs
Speaker: Theodore Molla, University of Illinois at Urbana-Champaign
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

A factor is a subgraph that contains every vertex of its host graph. Many important classical results give sufficient conditions for the existence of a specific type of factor. For example, in 1952 Dirac proved that if every vertex in a graph with at least three vertices is adjacent to at least half of the vertices, then the graph contains a Hamiltonian cycle, which is cycle that touches every vertex exactly once. In this talk, we will discuss recent extensions and analogues of classical results similar to Dirac's Theorem.

### Wednesday, February 1, 2017

Title: Cut-off for a class of hyperplane arrangement
Speaker: Evita Nestoridi, Princeton University
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

Consider a real hyperplane arrangement and let $$C$$ denote the collection of the occuring chambers. Bidigare, Hanlon and Rockmore introduced a Markov chain on $$C$$ which is a generalization of some card shuffling models used in computer science, biology and card games: the famous Tsetlin library used in dynamic file maintenance and cache maintenance and the riffle shuffles are two important examples of hyperplane walks. A strong stationary argument gives the upper bound for the separation distance mixing time for this Markov chain. A coupon collector argument will lead to the lower bound, which is discussed for the first time. I will try to explain both the geometric and the probabilistic techniques used in the problem.

### Monday, January 30, 2017

Title: Graph limits, entropy, and counting
Speaker: Andrew Uzzell, University of Nebraska–Lincoln
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

Extremal graph theory is the study of optimization problems on families of graphs. Given a class of graphs $$\mathcal{C}$$ and a parameter $$f$$, what is the maximum (or minimum) value of $$f$$ over $$\mathcal{C}$$, and which members of $$\mathcal{C}$$ achieve the extreme value? Somewhat surprisingly, it turns out that in some cases, such extremal results can both tell us how large $$\mathcal{C}$$ is and provide information about the typical structure of an element of $$\mathcal{C}$$. One can ask similar optimization and counting questions when $$\mathcal{C}$$ is a class of so-called multicolored graphs. (Examples of multicolored graphs include simple graphs, directed graphs, and multigraphs with bounded edge multiplicity.)

The theory of graph limits, developed over the last dozen years by Lovász and co-authors, provides an analytic framework for studying large graphs. Motivated by work of Hatami, Janson, and Szegedy on hereditary classes of graphs, we use graph limits to determine the asymptotic size and typical structure of hereditary classes of multicolored graphs. Our proofs combine counting arguments with analytic properties of the space of multicolored graph limits. This is joint work with Victor Falgas-Ravry, Svante Janson, and Johanna Strömberg.

### Monday, January 23, 2017

Title: The word problem in automaton groups
Speaker: Ievgen Bondarenko, Kyiv Taras Shevchenko University
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

In this talk we will discuss the complexity of the word problem in automaton groups and its connection to the minimization of automata iterations. I will present a class of automaton groups, for which the word problem can be solved in subexponential time. The relation with the growth of action graphs of automata will be discussed. I will show a unified approach to the complexity of the Hanoi Towers game on n pegs for $$n\ge 4$$, growth of ω-periodic graphs, and automata with polynomial activity growth.

### Monday, January 9, 2017

Title: Biquasiles and Dual Graph Diagrams
Speaker: Sam Nelson, Claremont McKenna College
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

Dual graph diagrams are an alternate way to present oriented knots and links in $$R^3$$. In this talk we will see how to turn dual graph Reidemeister moves into an algebraic structure known as biquasiles and use this structure to define new integer-valued counting invariants of oriented knots and links.